The Arduino.org description that you refer to is for the older board, now discontinued. It has different features, such as the SD slot and has a different procedure for updating microcode. The Arduino library interface is similar, as both inherit from the Stream class, but the implementations are different.

The Arduino.cc WiFi Shield 101 is covered on site: https://www.arduino.cc/en/Main/ArduinoWiFiShield101From this site you can examine the schematic and board layout which is quite different from the older board.

As it comes out of the box today, the Arduino.cc WiFi Shield 101 works (and works well) with the Arduino.cc Zero board. The old WiFi Shield works with the older boards such as the Arduino Uno, and may or may not work with the Zero.

WiFi Shield 101 is a newer radio chip with different microcode and different (mostly better) performance. It should be possible to move your software to the 101 Shield by changing to #include <WiFi101.h> and perhaps making hopefully minor changes to a sketch.